String A, String B String A is stretched between two clamps...

String $A$, String $B$ String $A$ is stretched between two clamps separated by distance $L .$ String $B$, with the same linear density and under the same tension as string $A$, is stretched between two clamps separated by distance $4 L$. Consider the first eight harmonics of string $B .$ Which, if any, has a resonant frequency that matches a resonant frequency of string $A$ ?

Key Concepts

Resonant Frequencies

Resonant frequencies are the specific frequencies at which a physical system, such as a stretched string, naturally vibrates. For a string fixed at both ends, these frequencies occur when the string supports standing wave patterns. Each resonant frequency corresponds to a particular mode of vibration, where the length of the string accommodates an integer number of half wavelengths.

Standing Waves

Standing waves are patterns of vibration that remain fixed in space while the energy oscillates in time. In a string fixed at both ends, standing waves form when the reflected waves interfere with the incident waves in such a way that certain points (nodes) appear to be stationary. These patterns determine the resonant frequencies which are integral multiples of the fundamental frequency.

Harmonics

Harmonics refer to the higher frequency modes of vibration in a system that occur as integer multiples of the fundamental frequency. The fundamental mode is the lowest frequency at which the string vibrates, while the subsequent harmonics (second, third, etc.) represent higher-order modes showing additional nodes and antinodes. Each harmonic represents a distinct, resonant vibrational state of the string.

Dependence on String Length

For strings under uniform tension with constant linear density, the resonant frequencies are inversely proportional to the length of the string. This means that when the length of the string is increased, the fundamental and all harmonic frequencies decrease. This principle is crucial for understanding how strings of different lengths, even under identical tension and density conditions, can have matched resonant frequencies when harmonics are appropriately aligned.

Boundary Conditions

Boundary conditions specify the constraints at the ends of the vibrating string, typically that the displacement is zero at the fixed ends. These conditions are vital because they determine the allowed waveforms (standing waves) on the string and lead to the quantization of the resonant frequencies, ensuring that only certain wavelengths, and hence frequencies, are possible.

Transcript

Please give Ace some feedback